Decomposing complete equipartite graphs into odd square-length cycles: Number of parts even

In this paper we show that the complete equipartite graph with n parts, each of size 2 k , decomposes into cycles of length λ 2 for any even n ≥ 4 , any integer k ≥ 3 and any odd λ such that 3 ≤ λ < 2 n k and λ divides k . As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph with an even number of parts into cycles of length p 2 , where p is prime. In proving our main result, we have also shown the following. Let λ ≥ 3 and n ≥ 4 be odd and even integers, respectively. Then there exists a decomposition of the λ -fold complete equipartite graph with n parts, each of size 2 k , into cycles of length λ if and only if λ < 2 k n . In particular, if we take the complete graph on 2 n vertices, remove a 1 -factor, then increase the multiplicity of each edge to λ , the resultant graph decomposes into cycles of length λ if and only if λ < 2 n .